logarithmic vernier

once-upon-a-time a potentially useful technique

Verniers are found in various constructions, on micrometers including straight linear,
rolling-linear with a geared dial running a linear track, screw-winding-linear where
the control turns a high-precision screw to advance straight linear calipers ...

The vernier auxiliary scale was added to the reticule on linear rules to improve
resolution of the finest graduation-- it relied on the mechanical stability, accuracy,
and visibility of the rule to provide more than first-readable delineation of precision.
It was a one-order-larger-than-minimum visible (to be readable) 10ths-of-9-10ths scale,
registering on every 9th-consecutive point of the 10ths-scale in a 1-of-10 coincidence
estimation of 10-vrs-9 (something akin to a linear moire pattern) -this coincidence
itself needed close scrutiny when visibly more than 1-of-10 appeared to touch
(consecutively) along the vernier-10-vrs-9-main scales and the best-or mean-of-2-or-3
was usually the best reading;- balancing helped anyway. (The vernier usually included
twice the ten graduations.)

But the logarithmic slide rule however-accurate was not readily adaptable to vernier,
as it was not linear: each successive graduation was narrower to the right and wider
to the left, and continued so along the rule: any matching 10-vrs-9 interval would
mismatch on the next adjacent; corresponding at few graduations on the main scale.-
However, it could, estimate nondecimally, a portion on the interval (the vernier needed
an out-of-ratio scale, to be comprehensive on all intervals):
It required using the vernier to set its own scale:

As the finest graduations were nearly linear (whence linear-interpolation was taught)
the application would have required matching the vernier scale to the two adjacent
best-or nearest-coincidences, counting the graduations between -that being the
proportion modulus- usually not convenient tenths: and varying locally, too.
The reading would have involved mentally rapidly estimating conversions to and from
decimal: a useful talent (eg. 5/8ths, about 0.6; conversely 0.3 is between 2-or-3/8ths).

(The vernier could-be straight linear, the log-scale, as only one digit was being
approximated,- but would need graduations not found anywhere on the measured scale.
Otherwise for measuring proportional scales, the vernier could be proportional, too,
as that produces nearly perfectly linear counts, -but again with uncommon graduations:
eg. slightly smaller than the smallest of the measured scale -or- slightly trapezoidal
with differing scales atop and below:- the vernier might be dots over its measured scale
graduations; an operator could count to farther coincidences for better precision.)

For illustration if you have a slide rule [or java-equivalent] handy:
set C-scale 4.5 to D-scale 5.5, and count the 0.05 graduations, left and right:
the scales realign best at C=4.30:D=5.25 and 4.70:5.75,
over a linear count span centered on 8 graduations on the C, and 10 on the D.
Now nudge the C-scale, part of one graduation: for example, in trying,
C=4.45:D=5.45 and 4.65:5.70 best-realign: a span of 4 on the C, and 5 on the D,
1-left and 4-right: which estimates the C=4.5 is at about D=5.51, accurate to 0.01:
a gain of 5× precision. (This illustration avoids perfect alignments to
infinite counts, and places where the graduation changes or differs not by doubling.)

It would have been an interesting adaption: The extant scales would have worked
similarly in some places, -where the moirelike match was several long,- but in
many places, it was non moirelike, and would have needed an additional scale.
It was most suitable to circular slide rules ... but computer programmers then-recently
had digital calculator programs handier.